Finite Group Databasebeta

← #35329621

Subgroups — #35329621

Order 21248.

1582 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581782C211yesmaximal, characteristic
2C2×C4×Dic83⋊C4C211yesmaximal, characteristic
3C2×#19127552C211yesmaximal, characteristic
4#24905715C212yesmaximal
5#24905714C212yesmaximal
6#24905698C211yesmaximal, characteristic
7#24905696C211yesmaximal, characteristic
8#24905695C211yesmaximal, characteristic
9Dic83⋊#95C212yesmaximal
10Dic83⋊#95C212yesmaximal
11C83×#1436C211yesmaximal, characteristic
12C2×#14581782—24no—
13C2×#14581782—24no—
14C2×#14581782C418yescomplemented
15C22×C4⋊Dic83C2211yescharacteristic
16C22×Dic83⋊C4C2212yes—
17C22×Dic83⋊C4C2211yescharacteristic
18C22×Dic83⋊C4C2211yescharacteristic
19C22×Dic83⋊C4C2211yescharacteristic
20C22×Dic83⋊C4C2211yescharacteristic
21C22×Dic83⋊C4C2212yes—
22C22×C4×Dic83C2212yes—
23C22×C4×Dic83C2211yescharacteristic
24C22×C4×Dic83C2211yescharacteristic
25C4×Dic83⋊C4—24no—
26#19127552—24no—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2211yescharacteristic
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2212yes—
33C2×C83⋊#95C2212yes—
34C2×C83⋊#95C2212yes—
35C22×C4⋊C4×C83C2211yescharacteristic
36C22×C42×C83C2211yescharacteristic
37C2×#95×C83C2212yes—
38C2×#95×C83C2211yescharacteristic
39C2×#95×C83C2212yes—
40C23×Dic83C2311yescharacteristic
41C23×Dic83C2311yescharacteristic
42C23×Dic83C2312yes—
43C23×Dic83C2312yes—
44C23×Dic83C2311yescharacteristic
45C23×Dic83C2311yescharacteristic
46#14581782—232no—
47C2×C4⋊Dic83—22no—
48C2×C4⋊Dic83—22no—
49C2×C4⋊Dic83C2×C414yes—
50C2×Dic83⋊C4—24no—
51C2×Dic83⋊C4—24no—
52C2×Dic83⋊C4—22no—
53C2×Dic83⋊C4—22no—
54C2×Dic83⋊C4—22no—
55C2×Dic83⋊C4—22no—
56C2×Dic83⋊C4—22no—
57C2×Dic83⋊C4—22no—
58C2×Dic83⋊C4—24no—
59C2×Dic83⋊C4—24no—
60C2×Dic83⋊C4—22no—
61C2×Dic83⋊C4—22no—
62C2×Dic83⋊C4—24no—
63C2×Dic83⋊C4C2×C414yes—
64C2×Dic83⋊C4C2×C414yes—
65C2×Dic83⋊C4C2×C414yes—
66C2×Dic83⋊C4C2×C414yes—
67C2×Dic83⋊C4Q818yes—
68C2×C4×Dic83—24no—
69C2×C4×Dic83—24no—
70C2×C4×Dic83—24no—
71C2×C4×Dic83—22no—
72C2×C4×Dic83—42no—
73C2×C4×Dic83C2×C414yes—
74C2×C4×Dic83D414yes—
75C2×C4×Dic83D414yes—
76C83⋊#95—24no—
77C83⋊#95—24no—
78C83⋊#95—24no—
79C83⋊#95—22no—
80C83⋊#95—24no—
81C83⋊#95—42no—
82C83⋊#95—42no—
83C83⋊#95—42no—
84C23×C4×C83C2311yescharacteristic
85C23×C4×C83C2311yescharacteristic
86C23×C4×C83C2312yes—
87C23×C4×C83C2312yes—
88C23×C4×C83C2311yescharacteristic
89C2×C4⋊C4×C83—22no—
90C2×C4⋊C4×C83—22no—
91C2×C4⋊C4×C83C2×C414yes—
92C2×C42×C83—22no—
93C2×C42×C83—42no—
94#95×C83—22no—
95#95×C83—24no—
96#95×C83—42no—
97C22×Dic83—22no—
98C22×Dic83—22no—
99C22×Dic83—22no—
100C22×Dic83—22no—
101C22×Dic83—21no—
102C22×Dic83—21no—
103C22×Dic83—21no—
104C22×Dic83—21no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83C22×C412yes—
120C22×Dic83C22×C412yes—
121C22×Dic83C22×C412yes—
122C22×Dic83C22×C412yes—
123C22×Dic83C2×D412yes—
124C22×Dic83C2×D412yes—
125C22×Dic83C2×D412yes—
126C22×Dic83C2×D412yes—
127C22×Dic83C2×Q814yes—
128C22×Dic83C2×Q814yes—
129C22×Dic83#4114yes—
130C22×Dic83#4114yes—
131C4⋊Dic83—28no—
132Dic83⋊C4—216no—
133Dic83⋊C4—28no—
134Dic83⋊C4—28no—
135Dic83⋊C4—28no—
136Dic83⋊C4—28no—
137Dic83⋊C4—216no—
138C4×Dic83—44no—
139C4×Dic83—48no—
140C4×Dic83C22⋊C4116yes—
141C24×C83C2411yescharacteristic
142C22×C4×C83—22no—
143C22×C4×C83—22no—
144C22×C4×C83—22no—
145C22×C4×C83—22no—
146C22×C4×C83—21no—
147C22×C4×C83—21no—
148C22×C4×C83—21no—
149C22×C4×C83—21no—
150C22×C4×C83—22no—
151C22×C4×C83—22no—
152C22×C4×C83—22no—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—22no—
164C22×C4×C83C22×C412yes—
165C22×C4×C83C22×C412yes—
166C22×C4×C83C22×C412yes—
167C22×C4×C83C2×D412yes—
168C22×C4×C83C2×D412yes—
169C22×C4×C83C2×Q814yes—
170C22×C4×C83#4114yes—
171C4⋊C4×C83—28no—
172C42×C83—44no—
173C23×C83C23×C411yescharacteristic
174C23×C83C22×D411yescharacteristic
175C23×C83C22×D411yescharacteristic
176C23×C83C22×Q812yes—
177C23×C83C2×#4112yes—
178C23×C83C2×#4112yes—
179C23×C83C2×#4112yes—
180C23×C83H(5,𝔽2)12yes—
181C23×C832−1+412yes—
182C2×C4×C83—24no—
183C2×C4×C83—22no—
184C2×C4×C83—22no—
185C2×C4×C83—24no—
186C2×C4×C83—22no—
187C2×C4×C83—22no—
188C2×C4×C83—22no—
189C2×C4×C83—22no—
190C2×C4×C83—24no—
191C2×C4×C83—24no—
192C2×C4×C83—24no—
193C2×C4×C83—24no—
194C2×C4×C83—24no—
195C2×C4×C83—24no—
196C2×C4×C83—24no—
197C2×C4×C83—42no—
198C2×C4×C83—42no—
199C2×C4×C83—42no—
200C2×C4×C83—42no—
201C2×C4×C83—42no—
202C2×C4×C83—41no—
203C2×C4×C83—41no—
204C2×C4×C83—42no—
205C2×C4×C83—42no—
206C2×C4×C83—42no—
207C2×C4×C83—42no—
208C2×C4×C83—42no—
209C2×C4×C83—42no—
210C2×C4×C83C2×C22⋊C414yes—
211C2×Dic83—22no—
212C2×Dic83—22no—
213C2×Dic83—24no—
214C2×Dic83—24no—
215C2×Dic83—22no—
216C2×Dic83—22no—
217C2×Dic83—22no—
218C2×Dic83—22no—
219C2×Dic83—24no—
220C2×Dic83—24no—
221C2×Dic83—24no—
222C2×Dic83—24no—
223C2×Dic83—24no—
224C2×Dic83—24no—
225C2×Dic83—24no—
226C2×Dic83—42no—
227C2×Dic83—42no—
228C2×Dic83—42no—
229C2×Dic83—42no—
230C2×Dic83—42no—
231C2×Dic83—41no—
232C2×Dic83—41no—
233C2×Dic83—42no—
234C2×Dic83—42no—
235C2×Dic83—42no—
236C2×Dic83—42no—
237C2×Dic83—42no—
238C2×Dic83—42no—
239C2×Dic83C2×C22⋊C414yes—
240C2×Dic83C2×C22⋊C414yes—
241C2×Dic83C4×Q818yes—
242C2×Dic83C22⋊Q818yes—
243C2×Dic83#12414yes—
244C2×Dic83C4⋊Q814yes—
245C22×C83C22×C22⋊C411yescharacteristic
246C22×C83C2×C4×Q812yes—
247C22×C83C4×#4111yescharacteristic
248C22×C83C4×#4111yescharacteristic
249C22×C83#52012yes—
250C22×C83C2×C22⋊Q812yes—
251C22×C83#52512yes—
252C22×C83C2×#12411yescharacteristic
253C22×C83C2×C4⋊Q811yescharacteristic
254C22×C83#53212yes—
255C22×C83#53511yescharacteristic
256C22×C83#53611yescharacteristic
257C22×C83#53712yes—
258C22×C83#55212yes—
259C22×C83#55212yes—
260C22×C83#55312yes—
261C22×C83#55312yes—
262C22×C83#55412yes—
263C22×C83#55412yes—
264C22×C83#55712yes—
265C22×C83#55712yes—
266C4×C83—24no—
267C4×C83—44no—
268C4×C83—44no—
269C4×C83—42no—
270C4×C83—42no—
271Dic83—24no—
272Dic83—44no—
273Dic83—44no—
274Dic83—42no—
275Dic83—42no—
276Dic83#39118yes—
277#14160—831nomaximal, Sylow
278C2×C83C2×#39111yescharacteristic
279C2×C83#229711yescharacteristic
280C2×C83#231111yescharacteristic
281C2×C83#237012yes—
282C2×C83#237012yes—
283C2×C83#256212yes—
284C2×C83#256512yes—
285C2×C83#256612yes—
286C2×C83#258412yes—
287C22×#125—831no—
288C2×C4×C4⋊C4—831no—
289C2×C4×C4⋊C4—831no—
290#1441—832no—
291#1441—832no—
292#1436—831no—
293#1436—831no—
294#1436—831no—
295#1436D8311yescharacteristic
296C4⋊#95—832no—
297C4⋊#95—832no—
298C83#1416011yescharacteristic, complemented, Sylow
299C2×#125—838no—
300C2×#125—1664no—
301C2×#125—1664no—
302C22×C4⋊C4—831no—
303C22×C4⋊C4—831no—
304C22×C4⋊C4—831no—
305C22×C4⋊C4—831no—
306C22×C4⋊C4—832no—
307C22×C4⋊C4—831no—
308C22×C4⋊C4—832no—
309C22×C4⋊C4C2×D8311yescharacteristic
310C22×C42—831no—
311C22×C42—831no—
312C22×C42—832no—
313C22×C42C2×D8311yescharacteristic
314C4×C4⋊C4—1664no—
315C4×C4⋊C4—1664no—
316C2×#95—832no—
317C2×#95—832no—
318C2×#95—832no—
319C2×#95—832no—
320C2×#95—832no—
321C2×#95—832no—
322C2×#95—831no—
323C2×#95—832no—
324C2×#95C2×D8312yes—
325C2×#95C2×D8311yescharacteristic
326C2×#95C2×D8312yes—
327C23×C4—831no—
328C23×C4—831no—
329C23×C4—831no—
330C23×C4—831no—
331C23×C4—832no—
332C23×C4—832no—
333C23×C4C22×D8311yescharacteristic
334C23×C4C22×D8311yescharacteristic
335C23×C4C22×D8312yes—
336C23×C4C22×D8312yes—
337C23×C4C22×D8311yescharacteristic
338#125—16632no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—834no—
342C2×C4⋊C4—834no—
343C2×C4⋊C4—834no—
344C2×C4⋊C4—834no—
345C2×C4⋊C4—834no—
346C2×C4⋊C4—838no—
347C2×C4⋊C4—1662no—
348C2×C4⋊C4—1662no—
349C2×C4⋊C4—1662no—
350C2×C4⋊C4—1662no—
351C2×C4⋊C4—1662no—
352C2×C4⋊C4—1662no—
353C2×C4⋊C4—1662no—
354C2×C4⋊C4—1662no—
355C2×C4⋊C4—1664no—
356C2×C4⋊C4—1664no—
357C2×C4⋊C4—1664no—
358C2×C4⋊C4—1664no—
359C2×C4⋊C4—1662no—
360C2×C4⋊C4—1662no—
361C2×C4⋊C4—1664no—
362C2×C4⋊C4C4×D8314yes—
363C2×C42—22no—
364C2×C42—42no—
365C2×C42—3322no—
366C2×C42—834no—
367C2×C42—834no—
368C2×C42—834no—
369C2×C42—1664no—
370C2×C42—1664no—
371C2×C42—1664no—
372C2×C42—1662no—
373#95—22no—
374#95—24no—
375#95—42no—
376#95—3322no—
377#95—3322no—
378#95—3322no—
379#95—1664no—
380#95—1664no—
381#95—1664no—
382#95—1662no—
383#95—1664no—
384C24C23×D8311yescharacteristic
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—832no—
408C22×C4—832no—
409C22×C4—832no—
410C22×C4—834no—
411C22×C4—834no—
412C22×C4—832no—
413C22×C4—832no—
414C22×C4—832no—
415C22×C4—832no—
416C22×C4—832no—
417C22×C4—834no—
418C22×C4—834no—
419C22×C4—1662no—
420C22×C4—1662no—
421C22×C4—1661no—
422C22×C4—1661no—
423C22×C4—1661no—
424C22×C4—1661no—
425C22×C4—1662no—
426C22×C4—1662no—
427C22×C4—1662no—
428C22×C4—1662no—
429C22×C4—1662no—
430C22×C4—1662no—
431C22×C4—1662no—
432C22×C4—1662no—
433C22×C4—1662no—
434C22×C4—1662no—
435C22×C4—1662no—
436C22×C4—1662no—
437C22×C4—1662no—
438C22×C4—1662no—
439C22×C4—1662no—
440C22×C4—1662no—
441C22×C4C2×C4×D8312yes—
442C22×C4C2×C4×D8312yes—
443C22×C4C2×C4×D8312yes—
444C22×C4D4×D8312yes—
445C22×C4D4×D8312yes—
446C22×C4Q8×D8314yes—
447C22×C4#1305059014yes—
448C4⋊C4—28no—
449C4⋊C4—1668no—
450C4⋊C4—1668no—
451C4⋊C4—1668no—
452C4⋊C4—1668no—
453C4⋊C4—16616no—
454C4⋊C4—16616no—
455C4⋊C4—1668no—
456C42—44no—
457C42—3324no—
458C42—3328no—
459C42—8316no—
460C23C22×C4×D8311yescharacteristic
461C23C2×D4×D8311yescharacteristic
462C23C2×D4×D8311yescharacteristic
463C23C2×Q8×D8312yes—
464C23#41×D8312yes—
465C23#41×D8312yes—
466C23C2×#1305059012yes—
467C23#1458183712yes—
468C23#1458183912yes—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—3322no—
498C2×C4—3322no—
499C2×C4—3322no—
500C2×C4—3322no—
501C2×C4—3322no—
502C2×C4—3322no—
503C2×C4—3322no—
504C2×C4—3322no—
505C2×C4—3322no—
506C2×C4—3321no—
507C2×C4—3321no—
508C2×C4—3322no—
509C2×C4—3322no—
510C2×C4—834no—
511C2×C4—838no—
512C2×C4—838no—
513C2×C4—834no—
514C2×C4—834no—
515C2×C4—834no—
516C2×C4—1664no—
517C2×C4—1664no—
518C2×C4—1662no—
519C2×C4—1662no—
520C2×C4—1664no—
521C2×C4—1664no—
522C2×C4—1662no—
523C2×C4—1662no—
524C2×C4—1664no—
525C2×C4—1664no—
526C2×C4—1664no—
527C2×C4—1662no—
528C2×C4—1662no—
529C2×C4—1664no—
530C2×C4—1664no—
531C2×C4C22⋊C4×D8314yes—
532C22C2×C22⋊C4×D8311yescharacteristic
533C22C4×Q8×D8312yes—
534C22#1912816911yescharacteristic
535C22C4×#1305059011yescharacteristic
536C22#1912818012yes—
537C22C22⋊Q8×D8312yes—
538C22#1912821012yes—
539C22#124×D8311yescharacteristic
540C22C4⋊Q8×D8311yescharacteristic
541C22#1912824812yes—
542C22#1912826611yescharacteristic
543C22#1912827311yescharacteristic
544C22#1912827812yes—
545C22#1912838412yes—
546C22#1912838712yes—
547C22#1912839612yes—
548C22#1912840012yes—
549C22#1912840512yes—
550C22#1912840612yes—
551C22#1912842812yes—
552C22#1912842912yes—
553C4—24no—
554C4—44no—
555C4—42no—
556C4—42no—
557C4—44no—
558C4—3324no—
559C4—3322no—
560C4—3322no—
561C4—3324no—
562C4—838no—
563C4—1664no—
564C2#391×D8311yescharacteristic
565C2#2490994911yescharacteristic
566C2#2491005011yescharacteristic
567C2#2491055412yes—
568C2#2491056012yes—
569C2#2491262712yes—
570C2#2491265212yes—
571C2#2491266112yes—
572C2#2491284912yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×#1436C2
socle, prime radical, cosocleC24×C83C24
hypercenter#1436D83
hypocenterC83#14160
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1436

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
14. C2×#14581782C4semidirect
298. C83#14160semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145817822
C2×C4×Dic83⋊C42
C2×#191275522
#249057152
#249057142
#249056982
#249056962
#249056952
Dic83⋊#952
C83×#14362
#1416083